Chapter 2.2, Problem 2.30E

To determine

To construct: the box plot on basis of given data and describe the distribution.

Explanation of Solution

Given:

3 0 12 10 8 4 2 7 10 11 6 7 0 35 12 3 10 2 7 9 15 5 14 15 6

Formula used:

  $\begin{array}{c}{\text{Q}}_1=1\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ {\text{Q}}_2=2\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ {\text{Q}}_3=3\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ {\text{IQR=Q}}_3-{\text{Q}}_1\end{array}$

  ${\text{Q}}_3+1.5\left(\text{IQR}\right)$

  ${\text{Q}}_1-1.5\left(\text{IQR}\right)$

Calculation:

Order the numbers in ascending order

0 0 2 2 3 3 4 5 6 6 7 7 7 8 9 10 10 10 11 12 12 14 15 15 35

  $\begin{array}{c}{\text{Q}}_1=1\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =1\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=6}\text{.5}\\ \text{=6th position +0}\text{.5}\left(7th-6th\right)\\ =3+0.5\left(4-3\right)\\ =3.5\end{array}$

  $\begin{array}{c}{\text{Q}}_2=2\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =2\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=13th postion}\\ {\text{M=Q}}_2=7\end{array}$

  $\begin{array}{c}{\text{Q}}_3=3\times \left(\frac{\text{N+1}}{4}\right)\text{th postion}\\ =3\times \left(\frac{\text{25+1}}{4}\right)\text{th postion}\\ \text{=19}\text{.5}\\ \text{=19th position+0}\text{.5}\left(20th-19th\right)\\ =11+0.5\left(12-11\right)\\ =11.5\\ {\text{Q}}_3=7\end{array}$

  $\begin{array}{c}{\text{IQR=Q}}_3-{\text{Q}}_1\\ =11.5-3.5\\ =8\end{array}$

For outliers

  $\begin{array}{c}{\text{Q}}_3+1.5\left(\text{IQR}\right)=11.5+1.5\left(8\right)\\ =23.5\\ {\text{Q}}_1-1.5\left(\text{IQR}\right)=3.5-1.5\left(8\right)\\ =-8.5\end{array}$

Graph:

  

By seeing the stem plot centre is at 7 and 1^st quartile and 3^rd quartile are 3.5 and 11.5 respectively. Inter quartile range is 8 and the minimum and maximum values are 0 and 35 respectively so the spread of the data is between 0 and 35 and the mode which is 10 because of highest frequency of the number. There is one outlier and distribution is positive skewed or right skewed which looking forth.